Gen. Math. Notes, Vol. 27, No. 1, March 2015, pp. 55-68
ISSN 2219-7184; Copyright © ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Numerical Study for Heat Transfer in
Symmetric Porous Channel with Expanding
or Contracting Walls and Slip Boundary
Condition
K. Raslan1 and Zain F. Abu Shaeer2
1,2
Department of Mathematics, Faculty of Science
Al-Azhar University, Egypt
1
E-mail: kamal_raslan@yahoo.com
2
E-mail: zainfathi@hotmail.com
(Received: 22-11-14 / Accepted: 1-2-15)
Abstract
The homotopy analysis method (HAM) is implemented to obtain the approximate
solutions of the nonlinear evolution equations in mathematical physics. The
results obtained by this method have a good agreement with one obtained. It
illustrates the validity and the great potential of the homotopy analysis method in
solving partial differential equations.
Keywords: HAM, Porous channel; Expanding and contracting walls Slip
boundary condition; Heat transfer.
1
Introduction
The flow in channels and in circular pipes with permeable walls has received
considerable attention in the past few years. The earliest work of steady flow
56
K. Raslan et al.
across permeable and stationary walls can be traced back to Berman [1], who
showed that the governing equations can be reduced to single fourth-order
nonlinear ordinary differential equation which includes permeation Reynolds
number ℛ , and associated solution can be obtained. Laminar flow studies in
porous pipes or channels with expanding or contracting walls have received
considerable attention due to their applications in biophysical flows. These
include the model of pulsating diaphragms, filtration, blood flow and artificial
dialysis, binary gas diffusion, the model of air and blood circulation in the
respiratory system. In order to simulate the peristaltic motion by successive wall
contractions and expansions. Uchida and Aoki [15] first examined the viscous
flow inside an impermeable tube with contracting cross sections. Ohki [20]
investigated the unsteady flow in a porous semi-infinite tube, whose elastic wall
had a varied length and a stable cross section. To simulate the laminar flowed in
cylindrical solid rocket motors, Goto and Uchida [4] analyzed the laminar
incompressible flow in a semi-infinite porous pipe, whose radius varied with time.
Bujurke et al. [3] obtained a series solution to the unsteady flow in a contracting
or expanding pipe. Majdalani et al. [6] obtained an exact similarity solution to the
viscous flow with small wall contractions or expansions and weakly permeability.
Dauenhauer and Majdalani [5] obtained a numerical solution and Majdalani and
Zhou [7] got both numerical and asymptotical solutions for moderate to large
Reynolds numbers. Srinivasacharya [14] obtained a numerical solution to the flow
and heat transfer of couple stress fluidin a porous channel with expanding and
contracting walls. Si et al. [13] obtained analytic solutionto the micro-polar-fluid
flow through a semi-porous channel with an expanding or contracting wall.
Dinarvand [10] studied viscous flow through slowly expanding or contracting
porous walls with lowseepage Reynolds number: a model for transport of
biological fluids through vessels.
No-slip condition was no longer valid at the permeable surface. Some of both
experimental and theoretical studies stated that slip could not be ruled out as a
significant element in the understanding of certain flow peculiarities [17].
Beavers, Joseph [2] reported mass efflux experimentsand proved the existence of
a non-zero tangential (slip) velocity on the surface of a permeableboundary. Using
a statistical approach, Saffman [12]derived a form for the slip velocity. Isenberg
[8] posited slip for all practical purposes in his study of blood flow in capillary
tubes. However, verylittle reports were found in literature for micropolar fluids
with expanding or contracting walls andslip boundary condition. Bennett [11]
reported that microscopic examination of blood flowing pasta glass wall shown
slipping (skidding) of red cells in contact with the wall. Chellam et al. [9]
investigated the effect on fluid flow and mass transfer with slip at a uniformly
porous boundary.
Recently, Zhang and Jia [18, 19] discussed the Navier-Stokes equations with firstorder and second-order accurate slip boundary conditions for describing the twodimensional gaseous steady laminar flow between two plates. Ramos [16]
obtained an asymptotic analytical solution of channel flows of incompressible
Numerical Study for Heat Transfer in…
57
fluids with a slip length that depended on the pressure and/or the axial
pressuregradient.
The series solutions are presented by HAM developed by Liao [21-23], which has
been successfully applied to several nonlinear problemsby Hayat, Noor and
Hashim [25, 26]. In this paper, the effects of different parameters, especially
expansion ratio and slip coefficient, on velocityand temperature are studied and
shown graphically.
The main goal of this paper is to find the numerical solutions for the heat transfer
insymmetric porous channel with expanding or contracting walls and slip
boundary conditions. Thesecond section will give statement of the problem and
governing equations. In section3, computations by Homotopy analysis method.
Finally, the graphs for velocity components and heat transfer presented for
different values of the fluid parameters are plotted and discussed.
2
Statement of the Problem and Governing Equations
Consider the unsteady two-dimensional motion of an incompressible fluid with
heat transfer in a porous semi-infinite channel with expanding or contracting walls
with slip boundary condition. The distance 2 (t) between the porous walls is
much smaller than the width and length of the channel. One end of the channel is
closed by a complicated solid membrane. Both walls have equal permeability
and expand or contract uniformaly at a time-depended rate ( ). As shown in
Fig. 1, a coordinate system may be chosen with the origin at the center of channel.
Take and y to be co-ordinate axes parallel and perpendicular to the channel
walls and assume u and v to be the velocity components in the and y directions
respectively and θ is the temperature. Under these assumptions, the governing
equations are expanded as follows [27].
+
"#
"-. #
0
/
$
"#
+%
0
/
$
=0
+%
+%
$
+&
0
/
+&
+&
' =−
' =−
,
)
)
' = * 14 # ' + #
+ *+ , %
+ *+ &
+
,
,
0
' 3 + 4+ , 5,
7
(1)
58
K. Raslan et al.
Figure1: Coordinate system and bulk fluid motion
A general infinitesimal group of transformations under which given partial
deferential equations are invariant [27], the equations (1) are a set of linear
deferential equations. Ref [27] completed transforms as
%=
89
8:
, & = − 8; ,
89
(2)
And the stream function takes the form, [24, 27]
<=
=(>), % =
?@
?
, & = −=,
(3)
These transforms change the second equation in (1) to,
AB C
A:B
+ α #y
AE C
A:E
+ 2 A:F ' + ℛ #G
AF C
AE C
A:E
− A: A:F ' = 0,
AC AF C
(4)
He Suggest that the form of temperature take as
5 = H( , >) = I(>) +
,
J(>),
(5)
During the assumption of the third and fourth equation in (1) become
KJ
K, =
K=
K,J
+
L
N(O
>
+
ℛ
=)
+
Q
S
T
−
2ℛ
J
U = 0,
M
P
R
P
K> ,
K>
K> ,
K>
,
?
?@ ,
F + LM 1(O > + ℛ P =) ? + 4QR #? ' 3 + 2 J = 0,
?F V
?V
(6)
With the boundary conditions
K , =(1)
K=(1)
= −X
,
K>
K> ,
=(1) = 1,
I(1) = 1,
J(1) = 0,
Numerical Study for Heat Transfer in…
?F @(Y)
? F
= 0, =(0) = 0, I(0) = 5, ,
Where ℛ =
and Ea =
Z[\
Re /f
J(0) = 0,(7)
is permeation Reynolds number, P_ =
Z[\
F
59
]
`ab
c
is Prandtl number
is Eckart number. The wall permeance or injection coefficient A is
defined as h = i, it is a measure of wall permeability. It will be started to solve
Z
the nonlinear equations (4) and (6) with the boundary conditions.
3
ℛ
Computations by Homotopy Analysis Method [21-23]
From the rule of solution expression and the boundary conditions (7) it is
straightforward to choose the following initial guesses.
=Y (>) =
, JY (>) =
(j klmn F )
,(olm k)
, and IY (>) = 5, + (1 + 5, )>,
(j klmn F )
,(olm k)
(8)
The linear operators are selected as
ℒo =
B , ℒ, =
?B @
?
F , and ℒ m =
?F q
?
?F V
,
? F
(9)
These operators satisfy the following properties:
ℒo (ro > m + r, > , + rm > + rs ) = 0, ℒ, (rt > + rj ) = 0, and
ℒm (ru > + rv ) = 0,
Where rw (x = 1 − 8) are the constants.
(10)
Upon making use of above definitions, we construct the zero-order deformation
problems
(1 − z)ℒo {= − =Y | = z ℏ ℵo {= , Ĵ |,
= € (1, z) = −X = €€ (1, z), = (1, z) = 1, = €€ (0, z) = 0, = (0, z) = 1,
(1 − z)ℒ, (Ĵ − JY ) = z ℏ ℵ, {= , Ĵ |,
Ĵ (1, z) = 0, Ĵ (0, z) = 1,
(1 − z)ℒm (• − •Y ) = z ℏ ℵm (= , J, I‚),
I‚(1, z) = 1, I‚(0, z) = 5, ,
(11)
(12)
(13)
(14)
(15)
(16)
60
K. Raslan et al.
ℵo {= , Ĵ | =
B@
B
ℵ, {= , Ĵ | =
2ℛP Ĵ (>, z)
0 , χ| =
ℵm {G
+ O #>
F q(
,))
F
)
@ ( ,))
E@(
,))
E
+2
F@(
,))
F
' + ℛP #
+ LM #(O > + ℛP = (>, z)'
E@(
= (>, z) −
,))
E
q( ,))
+ QR (
F@(
@ ( ,)) F @ ( ,))
F
'
(17)
) −
,)) ,
F
(18)
0(y, p)
∂, ζ‚(y, p)
∂ζ‚(y, p)
∂G
0 (y, p)'
+ P_ ‡#α y + ℛ G
+ 4Ea (
), ˆ + 2χ(y, p),
,
∂y
∂y
∂y
(19)
If z ∈ [0, 1] is an embedding parameter and ℏ are the nonzero auxiliary
parameters then the zeroth-order deformation problems can be constructed as [15]
respectively. Using Taylor’s theorem, we can write
•
= (>, z) = =Y (>) + ∑Ž
••o =• (>)z , =• (>) = •!
•
Ĵ (>, z) = Ĵ Y (>) + ∑Ž
••o Ĵ • (>)z , Ĵ • (>) =
,
o
‘@ (
,))
o
‘ q(
,))
•!
o
• ‚
‚
I‚(>, z) = I‚Y (>) + ∑Ž
••o I• (>)z , I• (>) = •!
)‘
‘V (
)‘
,
,))
)‘
,
(20)
(21)
(22)
The convergence of the two series is strongly dependent upon ℏ. Assume that ℏis
chosen so that the series (20-22) are convergent at z = 1. From Equations (2022), we have
=(>, z) = =Y (>) + ∑Ž
••o =• (>),
(23)
J(>, z) = JY (>) + ∑Ž
••o J• (>),
(24)
I(>, z) = IY (>) + ∑Ž
••o I• (>),
(25)
Differentiating Equations (14) and (16) m times with respect toz, then setting
z = 0 and finally dividing them by m! , we obtain the following m“” -order
deformation problems.
ℒo (=• (>) − •• =•no (>)) = ℏ ℜ@• (>),
€ (1)
€€ (1),
€€ (0)
=•
= −X =•
=• (1) = 0, =•
= 0, and =• (0) = 0,
€€€
€
€€
€€€
€€
ℜ@• (>) = =•no + O(> =•no
+ 2 =•no
) + ∑•no
—•Y ℛ P (=•n—no =— − =•n—no =— ),
(s)
ℒ, (J• (>) − •• J•no (>)) = ℏ ℜ• (>),
q
(26)
(27)
(28)
(29)
Numerical Study for Heat Transfer in…
J• (0) = 0, J• (1) = 0,
61
(30)
™
€
˜no
€
€
ℜ˜ (y) = χ€€
˜no + P_ (( α y)χ˜nšno + ℛ ∑š•Y (G˜nšno χ˜ − 2χ˜nšno Gš ) +
€€
€€
Ea G˜nšno Gš ),
(31)
ℒm (ζ˜ (y) − •˜ ζ˜no (y)) = ℏ ℜ˜ (y),
ζ˜ (0) = θ, , ζ˜ (1) = 1,
›
(32)
(33)
›
˜no
€
€
€
€
ℜ˜ (y) = ζ€€
˜no + P_ [( α y)ζ˜nšno + ℛ ∑š•Y (f˜nšno ζ˜ + 4Ea G˜nšno Gš ] +
(34)
2χ€€
˜no ,
Where
•• = •
0,
1,
ž ≤ 1, 7
ž > 1.
(35)
The general solutions of Equations (26), (29) and (32) are
∗ (>)
=• (>) = =•
+ ro + r, > + rm > , + rs > m ),
∗ (>)
J• (>) = J•
+ rt + rj >),
∗ (>)
I• (>) = I•
+ ru + rv >),
(36)
(37)
(38)
∗ (y), ∗ (y)
in which G˜
χ˜
and ζ∗˜ (y)denote the special solutions of Equations (26),
(29) and (32) and the integral constants C£ (i = 1 − 8) are determined by
employing the boundaryconditions (27),(30) and(33).In this way, it is easy to
solve the linear nonhomogeneous. Equations (26), (29) and (32) by using
Mathematicaone after the other in the order m = 1, 2, 3, . . ..
=(>) = (uujojYY(olm¦)§ (−3880800(−3 + > , − 6X)(1 + 3X)s − 55440(−1 +
> , )(1 + 3X), (ℛP (−2 + > , + > s + 3(−6 + > , + > s )X) + 21O(1 + 3X)(−1 −
7X + > , (1 + 3X)))ℏ + (−1 + > , )(−2772O(1 + 3X), (25> s O(1 + 3X), −
210(1 + 3X)(1 + 7X) − 2> , (1 + 3X)(−105 + 19O + 15(−21 + 8O)X) +
O(13 + 3X(52 + 285X))) + 6ℛP, (703 + 14> v (1 + 3X), − 7> j (1 +
3X), (53 + 110X) − > , (1 + 3X)(173 + 7X(71 + 330X)) − > s (1 +
3X)(173 + 7X(71 + 330X)) + X(11248 + 21X(2063 + 3630X))) −
77¬(1 + 3X)(65> j O(1 + 3X), + > , (1 + 3X)(360(1 + 3X) + O(227 +
681X + 3240X , )) + > s (1 + 3X)(360(1 + 3X) + O(389 + 15X(121 +
216X))) − 3(240(1 + 3X)(1 + 9X) + O(227 + X(3178 + 3X(3793 +
6480X))))))ℏ, )),
(39)
62
K. Raslan et al.
If we take ϕ → 0, we get
=(>) = S
n E
O,(
² §
,
+
m
−
to E
,
+ 3O #
§
sY
−
E
,Y
+
' + ℛP #
sY
¯
,vY
−
m E
,vY
+
osY
+ O(−
omℛi ° ±
,YojY
−
+ tjYY + ,t,YY − mmjYY)'³ + ℛP, (²,sYY − mmjY + o²jYY + oYuvYY − o,²mjYY) +
,vYY
² ¯
4
n ¯
oo,
² §
+
sYY
,,u E
,vYY
,,u
+
om
),
´´
µF °
m ¯
,vYY
um E
uYm
(40)
Numerical Results and Discussion
Our computations show that the series solutions converge in the whole region of
ywhenℏ¶ = ℏ· = −1,. This section deals with the graphics and the interpretation
of the dimensionless wall dilation rateα and the slip coefficientϕ, on the and y
components of the fluid velocity and heat transfer distributions. Table (1), figures
2(a) and 2(b) present the comparison of self-axial velocity u/x profiles between
the HAM solutions and Ref. [24] analytical results for α = ±0.5, X = 0.
Figures (3–5) show the effect of slip coefficient on the velocity components and
the temperature distributions. We can observe that the slip coefficient ϕ has
obvious influence on the velocity and the temperature. Fig.2 shows that the axial
velocity is a decreasing function of X near to the center. However, it is an
increasing function of ϕ near to the walls. However, with the increase in ϕ , the
influence of X on the velocity and the temperature becomes smaller. We can also
find that the radial velocity is a deceasing function ofX in fig. 3. Fig.4 show that
the effect of the slip coefficient ϕ on temperature θ, it is obvious that as ϕ
increasing the temperature θ decreasing.The influence of the wall expansion ratio
α on velocity component u/x is given in Figures (6–11) for fixed ϕ in case of
injectionℛ = 1 and suctionℛ = −1. With the expansion of the wall, the axial
velocity increases. The maximum of streamwise velocity lies at the center of the
channel whether α is positive and the lower near the wall; however, whether α is
negative (contracting wall), increasing contraction ratio leads to lower axial
velocity near the center, and the higher near the wall. The axial velocity
distribution, in all cases, approaches a cosine profile. Figs. (8 and 11) shows that
the effect of the wall expansion ratio α on the temperature θ, It is shown that the
temperature is increasing function with α for injection and suction. Figs. (12-14)
shows that the effect of the Eckert number Ea , Prandtel number P_ and the
initialtemperatureθ, on the temperature distribution, we find that the temperature
is increasing function withEa , P_ and θ, .
Numerical Study for Heat Transfer in…
63
Table1: Comparison between present work solutions and Ref. [24] for self-axial
velocity at ℛ = 5, forX = 0,O = −0.5 and O = 0.5
α = -0.5
α = 0.5,
Y
Ref[24]
Present
Percentage error
(%)
Ref[24]
Present
Percentage
error (%)
0.00
1.5151
1.50268
0.0124254
1.556324
1.5439
0.0124254
0.05
1.51134
1.49901
0.0123324
1.551780
1.53945
0.0123324
0.10
1.50005
1.488
0.0120513
1.538164
1.52611
0.0120513
0.15
1.48118
1.4696
0.0115757
1.515522
1.50395
0.0115757
0.20
1.45465
1.44376
0.0108947
1.483935
1.47304
0.0108947
0.25
1.42038
1.41039
0.00999335
1.443517
1.43352
0.00999335
0.30
1.37826
1.36941
0.00885233
1.394421
1.38557
0.00885233
0.35
1.32817
1.32072
0.00744968
1.336839
1.32939
0.00744968
0.40
1.27002
1.26426
0.00576282
1.271006
1.26524
0.00576282
0.45
1.20371
1.19994
0.00377264
1.197207
1.19343
0.00377264
0.50
1.1292
1.12773
0.00146993
1.115778
1.11431
0.00146993
0.55
1.04651
1.04764
0.00113512
1.027110
1.02824
0.00113512
0.60
0.955722
0.959722
0.00399957
0.931656
0.935656
0.00399957
0.65
0.857047
0.864077
0.00702959
0.829933
0.836962
0.00702959
0.70
0.750818
0.760875
0.0100568
0.722523
0.73258
0.0100568
0.75
0.650349
0.650349
0.0128079
0.610078
0.622886
0.0128079
0.80
0.532795
0.532795
0.0148669
0.493322
0.508189
0.0148669
0.85
0.408567
0.408567
0.0156289
0.373046
0.388675
0.0156289
0.90
0.278066
0.278066
0.014244
0.250109
0.264353
0.014244
0.95
0.141729
0.141729
0.00955104
0.125435
0.134986
0.00955104
1.00
-3.470E-18
0.00000
3.470E-18
3.470E-18
0.00000
3.470E-18
64
Fig (2a): Comparison between present work solutions and Ref.
[24] for self-axial velocity profiles at ¿ = 0.5 and ¼À = 5.0.
Fig. 3:The effect of the slip coefficient » on the selfaxial velocity component with ¼½ =0.5, ¾=0.2
Fig. 5: The effect of the slip coefficient » on temperture
distribution with ¼½ = 1, ¾ =0.5, ÂÃ = 0.5, ÄÅ = 0.7, ÆÇ =
0, È =3.
K. Raslan et al.
Fig (2b):Self-axial velocity profiles shown over a
range of Re at ¾ = −Á. » = Á
Fig.4: The effect of the slip coefficient » on the
radial velocity component with ¼½ =0.5, ¾=0.2
Fig. 6: The effect of the wall expansion ratio O onthe
self-axialvelocity component with ℛ =1, ϕ =0.2.
Numerical Study for Heat Transfer in…
Fig. 7: The effect of the wall expansion ratio O on the
radialvelocity component with ℛ = 1, ϕ=0.2.
Fig. 9: The effect of the wall expansion ratio Oonthe selfaxialvelocity component with ℛ = -0.5, ϕ =0.2.
Fig. 11: The effect of the wall expansion ratio Oon
temperaturedistribution with ℛ =- 1, X =0.2, Q- =
0.5,LË = 0.7, 5, = 0,x=3
65
Fig. 8: The effect of the wall expansion ratio O on
temperature distribution with ℛ = 1, X = 0.2,
Pr = 0.7, Ec=0.5, θ, = 0, x =3.
Fig. 10: The effect of the wall expansion ratio O on
the radial velocity component with ℛ = -0.5,
ϕ=0.2.
Fig. 12:The effect of the Eckart number QR on
temperaturedistribution with ℛ = 1, , X = 0.2α =0.5,
Pr = 0.7, θ, = 0, x =3.
66
K. Raslan et al.
Fig. 13:The effect of Prandle number ÄÅ on
temperature distribution with ¼½ = 0.5, Ì = Á. Ç, ¾
=0.5, ÂÃ = 0.5,ÆÇ = 0, È =3.
5
Fig. 14: The effect of ÆÇ on temperature
distribution with ¼½ = 0.5, » = Á. Ç, ¾ =0.5, ÂÃ =
0.5, ÄÅ = 0.7, È =3.
Conclusion
In this paper, the Homotopy Analysis Method has been applied to study the heat
transfer in symmetric porous channel with expanding or contracting walls and slip
boundary condition equation. The explicit series solutions our problem are
obtained, which are the same as those results given by Lie group analysis method
[27]for ℏ = −1. In conclusion, HAM provides accurate numerical solution for
nonlinear problems in comparison with other methods. It also does not require
large computer memory and discretization of the variables and>. The results
show that HAM is powerful mathematical tool for solving nonlinear partial
differential equations. Mathematica has been used for computations in this paper.
References
[1]
[2]
[3]
[4]
[5]
A.S. Berman, Laminar flow in channels with porous walls, J. of Appl.
Phys., 24(1953), 1232-1235.
G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally
permeable wall, Jour. of Fluid Mech., 30(1967), 197-207.
N.M. Bujurke, N.P. Pai and G. Jayaraman, Computer extended series
solution forum steady flowing a contracting or expanding pipe, IMA J. of
Appl. Math., 60(1998), 151165.
M. Goto and S. Uchida, Unsteady flow in a semi-infinite expanding pipe
with injection through wall, J. of the Japan Society for Aer. and Space Sc.,
33(1990), 14-27.
C.E. Dauenhauer and J. Majdalani, Exact self-similarity solution of the
Navier-Stokes equations for a porous channel with orthogonally moving
walls, Physics. of Fluids, 15(2003), 1485-1495.
Numerical Study for Heat Transfer in…
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
67
J. Majdalani, C. Zhou and C.D. Dawson, Two-dimensional viscous flow
between slowly expanding or contracting walls with weak permeability,
Jour. of Biomech., 35(2002), 1399-403.
J. Majdalani and C. Zhou, Moderate-to-large injection and suction driven
channel flows with expanding or contracting walls, ZAMM.Z Angew Math
Mech., 83(2003), 181-196.
I. Isenberg, A note on the flow of blood in capillary tubes, Bull. of Math.
Biology, 15(1953), 149-152.
S. Chellam, M.R. Wiesner and C. Dawson, Slip at a uniformly porous
boundary: Effect on fluid flow and mass transfer, Jour. of Eng. Math.,
26(1992), 481-492.
S. Dinarvand, Viscous flow through slowly expanding or contracting
porous walls with low seepage Reynolds number: A model for transport of
biological fluids through vessels, Com. Meth. In Biomech. and Biomed.
Eng., 14(2011), 853-862.
L.Bennet, Red cell slip at a wall in vitro, Science (Washington),
155(1967), 1554-1556.
P.G. Saffman, On the boundary condition at the surface of a porous
medium, Studiesin Appl. Math., 50(1971), 93-101.
X. Si, L. Zheng, X. Zhang and Y. Chao, Analytic solution to the
micropolar-fluid flow through a semi-porous channel with an expanding or
contracting wall, Appl. Math. Mech. Engl. Ed., 31(2010), 1073-1080.
D. Srinivasacharya, N. Srinivasacharyulu and O. Odelo, Flow and heat
transfer of couple stress fluid in a porous channel with expanding and
contracting walls, Int. Commun. Heat and Mass Transfer, 36(2009), 180185.
S. Uchida and H. Aoki, Unsteady flows in a semi-infinite contracting or
expanding pipe, J Fluid Mech., 82(1977), 371-387.
J.I. Ramos, Asymptotic analysis of channel flows with slip lengths that
depend on the pressure, Appl. Math. and Computation, 188(2007), 13101318.
V and V, Viscosity of solutions and suspensions, Jour. of Phys. and
Colloid Chem., 52(1948), 277-299.
T.T. Zhang, L. Jia, Z.C. Wang and X. Li, The application of homotopy
analysis method for 2-dimensional steady slip flow in micro channels,
Phys. Lett. A, 372(2008), 3223-3227.
T.T. Zhang, L. Jia and Z.C. Wang, Validation of Navier–Stokes equations
for slip flow analysis within transition region, Int J Heat Mass Trans.,
51(2008), 6323-6327.
M. Ohki, Unsteady flows in a porous, elastic, circular tube- Part 1: The
wall contracting or expanding in an axial direction, Bulletin of the JSME,
23(1980), 679-686.
S.J. Liao, Notes on the homotopy analysis method: Some definitions and
theorems, Comm. Non-Linear Sci. Num. Simul., 14(2009), 983-997.
S.J. Liao, On the relationship between the homotopy analysis method and
Euler transform, Comm. Nonlinear Sci. Numer. Simulat., 15(2010), 14211431.
68
[23]
[24]
[25]
[26]
[27]
K. Raslan et al.
S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl
Math Comput., 147(2004), 499-513.
Y.Z. Boutros, M.B. Abd-el-Malek, N.A. Badran and H.S. Hassan, Liegroup method solution for two-dimensional viscous flow between slowly
expanding or contracting walls with weak permeability, Appl. Math.
Modelling, 31(2007), 1092-1108.
T. Hayat and M. Khan, Homotopy solution for a generalized second grade
fluid past aporous plate, Nonlinear Dyn., 42(2005), 395-405.
N.F.M. Noor and I. Hashim, Thermo capillarity and magnetic field effects
in a thin liquid film on an unsteady stretching surface, Int. J. Heat Mass
Trans., 53(2010), 2044-2051.
Kh. S. Mekheimer, M. El-Sabbagh and R.E. Abo-Elkhair, Lie group
analysis for the heat transfer in symmetric porous channel with expanding
or contracting walls and slip boundary condition, http://www.etms-eg.org.